one more refs and minor corrections

CCvsMBPT.bib

@@ -1,13 +1,30 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2022-10-12 10:53:04 +0200 
+%% Created for Pierre-Francois Loos at 2022-10-12 13:42:59 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@inbook{Bartlett_1986,
+	abstract = {A diagrammatic derivation of the coupled-cluster (CC) linear response equations for gradients is presented. MBPT approximations emerge as low-order iterations of the CC equations. In CC theory a knowledge of the change in cluster amplitudes with displacement is required, which would not be necessary if the coefficients were variationally optimum, as in the CI approach. However, it is shown that the CC linear response equations can be put in a form where there is no more difficulty in evaluating CC gradients than in the variational CI procedure. This offers a powerful approach for identifying critical points on energy surfaces and in evaluating other properties than the energy.},
+	address = {Dordrecht},
+	author = {Bartlett, Rodney J.},
+	booktitle = {Geometrical Derivatives of Energy Surfaces and Molecular Properties},
+	date-added = {2022-10-12 13:42:54 +0200},
+	date-modified = {2022-10-12 13:42:59 +0200},
+	doi = {10.1007/978-94-009-4584-5_4},
+	editor = {J{\o}rgensen, Poul and Simons, Jack},
+	isbn = {978-94-009-4584-5},
+	pages = {35--61},
+	publisher = {Springer Netherlands},
+	title = {Analytical Evaluation of Gradients in Coupled-Cluster and Many-Body Perturbation Theory},
+	url = {https://doi.org/10.1007/978-94-009-4584-5_4},
+	year = {1986},
+	bdsk-url-1 = {https://doi.org/10.1007/978-94-009-4584-5_4}}
+
 @article{Forster_2020,
 	author = {F{\"o}rster, Arno and Visscher, Lucas},
 	date-added = {2022-10-12 10:52:33 +0200},

CCvsMBPT.tex

@@ -457,7 +457,7 @@ Partial self-consistency can be attained via the \textit{``eigenvalue''} self-co
 In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation 
 \begin{equation}
 	\label{eq:GW}
-	\qty[ \be{}{} + \bSig{}{\GW}\qty(\e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
+	\qty[ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
 \end{equation}
 where $\be{}{}$ is a diagonal matrix gathering the HF orbital energies and the elements of the correlation part of the dynamical (and non-hermitian) $GW$ self-energy are
 \begin{equation}
@@ -493,7 +493,7 @@ As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the frequency-dependent
 	\cdot
 	\be{}{\GW}
 \end{equation}
-where $\be{}{\GW}$ is a diagonal matrix gathering the quasiparticle energies, the 2h1p and 2p1h matrix elements are
+where $\be{}{\GW}$ is a diagonal matrix collecting the quasiparticle energies, the 2h1p and 2p1h matrix elements are
 \begin{subequations}
 \begin{align}
 	C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \e{i}{\GW} + \e{j}{\GW} - \e{a}{\GW}) \delta_{jl} \delta_{ac} - \ERI{jc}{al} ] \delta_{ik} 
@@ -510,7 +510,7 @@ and the corresponding coupling blocks read
 Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}
 
 Let us suppose that we are looking for the $N$ \textit{``principal''} (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}. 
-Therefore, $\bX{}{}$ and $\be{}{\GW}$ are of size $N \times N$.
+Therefore, $\bX{}{}$ and $\be{}{\GW}$ are square matrices of size $N \times N$.
 Assuming the existence of $\bX{}{-1}$ and introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have
 \begin{equation}
 	\begin{pmatrix}
@@ -613,7 +613,7 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $
 	\bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} 
 \end{equation}
 Again, similarly to the dynamical equations defined in Eq.~\eqref{eq:GW} which requires the diagonalization of the dRPA eigenproblem [see Eq.~\eqref{eq:dRPA}], the CC equations reported in Eqs.~\eqref{eq:r_2h1p} and \eqref{eq:r_2p1h} can be solved with $\order*{N^6}$ cost by defining judicious intermediates.
-Cholesky decomposition, density fitting, and other related techniques might be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
+Cholesky decomposition, density fitting, and other related techniques may be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
 The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equations for each value of $p$ separately.
 
 %%%%%%%%%%%%%%%%%%%%%
@@ -625,7 +625,7 @@ The conventional and CC-based versions of the BSE and $GW$ schemes that we have
 Similitudes between BSE@$GW$ and STEOM-CC have been also highlighted, and may explain the reliability of BSE@$GW$ for the computation of optical excitations in molecular systems. 
 
 We hope that the present work may provide a consistent approach for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} and BSE \cite{IsmailBeigi_2003,Caylak_2021,Knysh_2022} frameworks, hence broadening the applicability of these formalisms in computational photochemistry.
-However, several challenges lie ahead as one must derive, for example, the associated $\Lambda$ equations and the response of the static screening with respect to the external perturbation at the BSE level.
+However, several challenges lie ahead as one must derive, for example, the associated $\Lambda$ equations \cite{Bartlett_1986} and the response of the static screening with respect to the external perturbation at the BSE level.
 The present connections between CC and $GW$ could also provide new directions for the development of multireference $GW$ methods \cite{Brouder_2009,Linner_2019} in order to treat strongly correlated systems. \cite{Lyakh_2012,Evangelista_2018}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%